Explanation Given: The height of the beam’s cross-sectional area is 450 mm . The width of the beam’s cross-sectional area is 400 mm . Show the beam’s cross-sectional area as in Figure (1). From Figure (1), As the beam’s cross-sectional are is symmetrical about the y axis, write the product of inertia for the shaded area with respect to the x and y axes as 0 . I x y = 0 Compute the product of inertia for the beam’s cross-sectional area with respect to the x axis. I x = 1 36 ( 2 a ) ( b + c ) 3 (I) Compute the product of inertia for the beam’s cross-sectional area with respect to the y axis. I y = 2 [ 1 36 ( b + c ) ( a ) 3 + 1 2 ( b + c ) ( a ) { a 3 } 3 ] (II) Compute the moment of inertia of the beam’s cross-sectional area with respect to the u axis. I u = [ I x + I y 2 ] + [ I x − I y 2 ] cos 2 θ − I x y sin 2 θ (III) Compute the moment of inertia of the beam’s cross-sectional area with respect to the v axis. I v = [ I x + I y 2 ] − [ I x − I y 2 ] cos 2 θ + I x y sin 2 θ (IV) Express the product of inertia of the beam’s cross-sectional area with respect to the v axis. I u v = I x − I y 2 sin 2 θ + I x y cos 2 θ (V) Conclusion: Substitute 200 mm for a , 300 mm for c , and 150 mm for b in Equation (I). I x = 1 36 ( 2 × 200 ) ( 150 + 300 ) 3 = 1 36 ( 400 ) ( 450 ) 3 = 1 , 012.5 ( 10 6 ) mm 4 Substitute 200 mm for a , 300 mm for c , and 150 mm for b in Equation (II). I y = 2 [ 1 36 ( 150 + 300 ) ( 200 ) 3 + 1 2 ( 150 + 300 ) ( 200 ) { 200 3 } 3 ] = 2 [ 1 36 ( 450 ) ( 200 ) 3 + 1 2 ( 450 ) ( 200 ) ( 66.667 ) 3 ] = 600 ( 10 6 ) mm 4 Substitute 1 , 012.5 ( 10 6 ) mm 4 for I x , 600 ( 10 6 ) mm 4 for I y , 30 ° for θ , and 0 for I x y in Equation (III)

Engineering Mechanics: Statics and Study Pack (13th Edition)

13th Edition

ISBN: 9780133027990

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Question

Chapter 10.7, Problem 70P

To determine

The moments of inertia with respect to the u and v axes and the product of the beam’s cross-sectional area.

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Chapter 10.7, Problem 69P

Chapter 10.7, Problem 71P

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Chapter 10 Solutions

Engineering Mechanics: Statics and Study Pack (13th Edition)

Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Prob. 1P Ch. 10.3 - Prob. 2P Ch. 10.3 - Prob. 3P Ch. 10.3 - Prob. 4P Ch. 10.3 - Prob. 5P Ch. 10.3 - Prob. 6P

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